Chapter 9

Conceptual Models and Applied Mathematical
Problem-Solving Research*

Richard Lesh, Marsha Landau,
and Eric Hamilton

This chapter defines and illustrates a theoretical construct, the conceptual model, as an adaptive structure central to research at the interface of two NSF (National Science Foundation)-funded projects. The Rational Number (RN) project (Behr, Lesh. & Post, Note 1) investigates the nature of children's rational-number ideas in Grades 2-8. The Applied Problem Solving (APS) project (Lesh, Note 2) investigates successful and unsuccessful problem-solving behaviors of average-ability students, working on problems that involve easy to identify substantive mathematical content, and on realistic problem-solving situations, in which a variety of outside resources (including calculators, resource books, other students, and teacher-consultants) are available. Lesh (1981) gives a rationale for these emphases. The kinds of problems we focus on in this chapter are much "smaller" than those that have received our greatest attention in the APS project, more closely resembling the "word problems" that appear in elementary and middle school textbooks.

The chapter consists of three major sections. The first section particularizes the discussion of conceptual models by referring specifically to rational-number concepts.

The second section discusses results from interviews in which 80 fourth through eighth graders solved problems presented in a number of formats. The interviews revealed that: (a) during the solution process, subjects frequently change the problem representation from one form to another (e.g., written symbols to spoken words, spoken words to pictures or concrete models, etc); and, (b) at any given stage, two or more representational systems may be used simultaneously, each illuminating some aspects of the situation while de-emphasizing or distorting others. A major conclusion drawn from the data is that purportedly realistic word-problems often differ significantly from their real-world counterparts in their difficulty, the processes most often used in solutions, and the types of errors that occur.

The third section of the chapter presents results from the written testing program of the RN project (see Chapter 4). Item difficulty depended on the structural complexity of the underlying conceptual models, and on the types of representational translations that the item required. Baseline information was derived comparing fourth through eighth graders' abilities to translate within and between representational modes (written language, written symbols, pictures) on sets of structurally related tasks.

The chapter concludes with remarks about how the reported research advances our understanding of the growth and development of children's conceptual models and contributes to the theoretical framework for current work on the RN and APS projects.

Conceptual Models

A conceptual model is defined as an adaptive structure consisting of (a) within-concept networks of relations and operations that the student must coordinate in order to make judgments concerning the concept; (b) between-concept systems that link and/or combine within-concept networks; (c) systems of representations (e.g., written symbols, pictures, and concrete materials), together with coordinated systems of translations among and transformations within modes; and (d) systems of modeling processes; that is, dynamic mechanisms that enable the first three components to be used, or to be modified or adapted to fit real situations

For a given mathematical concept, the first two components of a student's conceptual model make up what might be called the student's understanding of the idea; within- and between-concept systems define the underlying structure of the concept. The third component includes a variety of qualitatively different systems for representing these understandings–using written symbols, spoken language, static figural models (e.g., pictures, graphs, diagrams), manipulative models (e.g., concrete materials), or real-world ''scripts." The fourth component contains processes for (a) changing the real situation to fit existing understandings, (b) changing existing understandings to fit the situation, and (c) changing the model to fill gaps, eliminate internal inconsistencies, and resolve conflicts within the model itself.

Representational systems differ from one another because they emphasize or de-emphasize different aspects of the underlying structure of the concept. They also differ in generative power and in their ability to manipulate relevant ideas and data simply and economically in various situations. For example, sometimes a picture is worth a thousand words; sometimes language is clearer and more efficient.

The distinction between understandings and representations of understandings is quite important in mathematics. Some major advances in mathematics have resulted from the creation of clever or powerful representations (e.g., Cartesian coordinates and decimal notation) that initially functioned primarily as externalized models of ideas (i.e., structures) that were already known. Later these representations provided new tools for generating new ideas.

The first three components of a conceptual model contain most of the ''actions" commonly associated with condition-action pairs in computer-simulated information-processing models of cognition. These actions transform information within the model, but their application does not lead to the development of new, more refined, or higher order conceptual models. Conceptual models are closed (in a mathematical sense) under the operations in the first three components. The fourth component consists of dynamic mechanisms that enable the first three components to develop and adapt to everyday applications.

Although between-concept systems (part b of the definition) and modeling processes (part d) are the components of conceptual models that are most salient for the types of problems that have been the major focus of the APS project, an adequate description of those components depends on a firm understanding of within-concept networks (part a) and systems of representations (part c), which are priority foci of the RN project. Therefore, this chapter focuses on parts a and c. Detailed treatments of parts b and d will appear in future publications from the APS project.

Background for Current Research
on Applied Problem-Solving

The APS project is unusual among problem-solving projects in mathematics education because, rather than emanating from instructional development or research on problem solving itself, it grew out of research on concept formation. The goals of that research included tracing the development of selected mathematical ideas and identifying task characteristics that influenced students' abilities to use the ideas in particular situations. Many of our theoretical perspectives evolved during investigations of mathematical abilities that are deficient in "learning disabilities'' subjects (Lesh, 1979b), social and affective factors that influence problem-solving behavior (Cardone, 1977; Lesh, 1979a), the development of spatial and geometric concepts in children and adults (Lesh & Mierkiewicz, 1978), and the role of representational systems in the acquisition and use of rational-number concepts (see Chapter 4 of this volume).

A central question that the APS project is designed to address is, ''What is it, beyond having an idea, that enables an average ability student to use it in realistic everyday situations?'' We believe that many of the most important applied problem-solving processes contribute significantly to both the meaningfulness and the usability of mathematical ideas. It is not necessarily the case that students first learn an idea, then add some general problem-solving processes, and finally (if ever) use the idea and processes in real situations, that is, those in which some knowledge about the situation is needed to supplement the underlying mathematical ideas and processes. Rather, there is a dynamic interaction between the content of mathematical ideas and the processes used to solve problems based on those ideas. This assumption has both practical and theoretical implications. Applications and problem solving are unlikely to be fully accepted in the school mathematics curriculum unless teachers and other practitioners are convinced that they play an important role in the acquisition of basic mathematical ideas. We believe that applications and problem solving should not be reserved for consideration only after learning has occurred; they can and should be used as a context within which the learning of mathematical ideas takes place.

The task of selecting or designing problems to use in research on applied mathematical problem-solving can be approached from at least three directions. First, one can start with important elementary mathematical ideas, sort out the different interpretations that these ideas can have in realistic situations, and identify problem situations in which these interpretations occur. This is the perspective of a recent project headed by Usiskin (Note 4) and Bell as well as parts of our RN project. One conclusion from these projects is that textbook word-problems typically represent only a narrow sample of idea interpretations and problem types that should be addressed.

A second perspective, and the one that has characterized the largest portion of our efforts on the APS project, starts with realistic everyday situations in which mathematics is used and attempts to identify the processes, skills, and understandings that are most important in their solution. Again, one conclusion is that the problem types, ideas, and skills that appear to be most critical are quite different from those that have been emphasized by mathematics education spokespersons for "basic skills" or ''problem solving," and by research on textbook word-problems (Lesh, 1983).

A third approach, which is the one taken in the second section of this chapter, is to start with typical textbook word-problems and create concrete or real-world situations characterized by the same mathematical structures. From this perspective, the goal is not to investigate responses to word problems as an end in itself; rather, the goal is to understand problem solving in realistic everyday situations.

Although an abundance of research has been conducted related to problem solving (see, for example, the literature review in Lester, Chapter 8 of this volume), there has been very little applied problem-solving research that investigates the development of conceptual models or that incorporates any of the three approaches to task selection described above.

Before describing the concrete or realistic word-problem isomorphs used in the interviews reported in the second section of this chapter, we first characterize 18 realistic problems, not based on word problems, which have provided the research sites for most of the investigations in the APS project.

Most of our problems have been designed to require 10-45 minutes for solution by average-ability seventh-graders, and to resemble problem-solving situations that might reasonably occur in the everyday lives of the students or their families. Realistic ''outside" resources were available, including other students, calculators, resource books and materials, and teacher consultants. Solution attempts were therefore not blocked by deficient technical skills (e.g., computation) or memory capabilities (e.g., recall of measurement facts).

All our problems were based on straightforward uses of easy-to-identify concepts from arithmetic, measurement, or intuitive geometry. No tricks were needed; the most direct solution path was a correct path, although it was not necessarily the most efficient or elegant.

In contrast with simple, one-step word-problems, a variety of solutions and solution paths were possible, varying in complexity and sophistication. The relevant ideas seldom fit into neat disciplinary categories; most of our problems required the retrieval and integration of ideas and procedures associated with a number of distinct topics, including several arithmetic operations, various number systems (e.g., rational numbers, negative numbers, decimals), several qualitatively distinct measurement systems (e.g., length, time), or intuitive ideas from geometry, physics, or other subject-matter areas.

Many of our problems were designed so that the critical solution stages would be ''nonanswer giving" stages. For example, in many realistic problem situations, problem formulation or trial answer refinement are crucial in the solution process. Thus, for many of our problems, the goal was not to produce a numerical "answer;" instead it was to make nonmathematical decisions, comparisons, or evaluations using mathematics as a tool.

In typical textbook word-problems, only two or three numbers usually appear, and the most common errors result when youngsters use one of their "number crunching routines" to produce an answer, sensible or not. When "too much" information is given, (for example, three numbers may appear instead of two) the student is expected to ignore the irrelevant information. When "not enough'' is given, the student is expected to conclude that the problem cannot be done. In more realistic situations, and in many of the problems that were the foci of our APS research, there was simultaneously ''too much'' and "not enough'' information. Often, there was an overwhelming amount of information, all relevant, and the main difficulty was to select and organize the information that was "most useful'' in order to find an answer that was "good enough.'' Furthermore, the given information may have consisted of both qualitative and quantitative information that had to be combined in some sensible way. In other problems, not enough information may have been provided, but a usable answer still had to be found. It may have been necessary to identify or generate additional information or data as part of a solution attempt when not all the information was given initially. Thus, in most of our problems, conceptual models serve as "filters" to select information from real situations, and as "interpreters" to organize or transform data, or to fill in (or compensate for) missing information. In such situations, a model always distorts or de-emphasizes some aspects of the real situation in order to clarify or emphasize others and a major goal of our APS research is to model students' modeling behaviors (Lesh, 1982).

In much the same way that our attempts to understand students' problem behaviors are embodied in the creation of models of their cognitive processing, we assume that our subjects' attempts to understand the problems we pose are embodied in the creation and adaptation of conceptual models. Below, each of the four components of a conceptual model (within-concept networks, between-concept systems, representational systems, and modeling mechanisms) is reconsidered in greater detail, with brief references to past research.

Within-Concept Networks

To most mathematicians, mathematics is the study of structure, the content of mathematics consists of structures, and to do mathematics is to create and manipulate structures. It is our hypothesis that these structures (whether they are embedded in pictures, manipulative materials, spoken language, or written symbols), and the processes used in manipulating and creating these structures, comprise the ''conceptual models'' that mathematicians and mathematics students use to solve problems.

In past research on the development of number and measurement concepts (Lesh, 1976), spatial and geometric concepts (Lesh & Mierkiewicz, 1978), and rational-number concepts (Lesh, Landau, & Hamilton, 1980), the similarities and differences between formal axiomatic structures and children's cognitive structures have been investigated. Formal axiomatic systems were used to generate tasks and clinical interview questions to help identify the nature of children's primitive conceptualizations of mathematical ideas. For example, the relationships that a child uses to think about and compare ratios are similar to the relations that define a ''complete ordered field' in which the elements are equivalence classes of ordered pairs of whole numbers. On the other hand, when a rational number is interpreted as a position on a number line, many of the relationships that children notice and use are simplified and restricted versions of the structural properties that define the metric topology of the rational-number line–such properties as betweenness, density, distance, and (non)completeness. According to the "number line'' interpretation the set of rational numbers is regarded (intuitively at first) as a subset of the set of real numbers, whereas, using the ''ratio'' interpretation, rational numbers are thought of as extensions of the whole numbers. A youngster's rational-number conceptual model has a within-concept network associated with each rational-number subconstruct, for example, ratio, number line, part-whole, operator, rate, and indicated quotient.

Between-Concept Systems

One of the most important properties of mathematical ideas, whether they occur as conceptual models used by children or as formal systems used by mathematicians, is that they are embedded in well-organized systems of ideas, so that part of the meaning of individual ideas derives from relationships with other ideas in the system or from properties of the system as a whole (Lesh, 1979b). In rational-number conceptual models, these between-concept systems have three components: (a) within-concept networks associated with different rational-number types (e.g., part-whole fractions, ratios, rates, decimals, and operator transformations); (b) links between those networks (including understandings of the "sameness'' and/or ''distinctness'' of the rational-number types); and (c) operations that, among other things, enable the transformation of a given rational number into different forms. Further, between-concept systems link rational-number ideas with other concepts such as measurement, whole-number division, and intuitive geometry concepts related to areas and number lines.

For most youngsters, between-concept systems associated with rational numbers are poorly organized and unevenly formalized. The between-concept systems derive some of their meaning from within-concept networks, and, in turn, the within-concept networks derive some of their meaning from the between-concept systems in which they are embedded. For example, the transformation of a simple fraction to a percent gives the fraction a meaning that includes proportion ideas.

That a whole structure and its parts each derive some meaning from the other has profound implications for human learning, development, and problem solving. The resulting chicken-and-egg dilemma concerning the ''whole structure'' versus "parts within the whole'' dictates that cognitive growth must involve more than quantitative additions to knowledge or processing capabilities– qualitative reorganizations must also occur. That is, as various ideas, relationships, and operations evolve into a whole system with properties of its own, elements within the system achieve a new status by being treated as parts of the whole.

Representational Systems

An introduction to the role of representational systems in applied problem solving is given in Lesh (1981), which includes a discussion of translation processes that contribute to the meaningfulness of mathematical ideas. Figure 9.1 shows some of the translations among different representational modes and some of the contributions to understanding ideas that those translations can make.

Though this depiction of modes and translations is neither exclusive nor exhaustive, it has suggested useful areas of investigation for the RN and the APS projects. Just as mathematical ideas are embedded in larger between-concept systems that contribute to the meaningfulness of those ideas, so are those ideas and between-concept systems embedded in representational systems such as those shown.

When we say a student understands a mathematical concept, part of what we mean is that he or she can use the kinds of translation processes depicted in Figure 9.1. For example, when we say a student understands fractions, we mean, in part, that he or she can express fraction ideas presented with circular regions using rectangular regions, or using written symbols.

Modeling Mechanisms

In applied problem-solving, important translation and/or modeling processes include (a) simplifying the original problem situation by ignoring "irrelevant" characteristics in a real situation in order to focus on other characteristics; (b) establishing a mapping between the problem situation and the conceptual model(s) used to solve the problem; (c) investigating the properties of the model in order to generate information about the original situation; and (d) translating (or mapping) the predictions from the model back into the original situation and checking whether the results ''fit.''

As the results we present in this chapter show, Figure 9.2 represents a useful, though oversimplified, conceptualization of the problem-solving process. For example, different aspects of the problem may be represented using different representational systems, and the solution process may involve mapping back and forth among several systems–perhaps using pictures as an intermediary between the real situation and written symbols.

In the next two sections of this chapter, we report data relating to two of the four aspects of conceptual models, namely within-concept networks and representational systems. Our goal is to lay groundwork for future data presentations and elaborations relating to between-concept systems and modeling mechanisms.

Rational-Number Task Interviews

This section reports the results of interviews designed to explore the differences in student responses to sets of problems based on the same arithmetic structures but varying in presentation format. Standard word-problems were included in the interview to permit the direct comparison of students' responses to word problems with their responses to corresponding real or more realistic problems.

Some generalizations emerged from the interviews taken as a whole; other important conclusions were found by examining in detail the results of individual problems. The chief findings were that (a) word problems differ from their real-world counterparts with respect to difficulty, the predominant representational mode selected for solution, and most frequent error types; (b) varying the tasks on any of a number of dimensions (e.g., number size, context, type of manipulative material present) is accompanied by variations in the subjects' performance, suggesting that most of our subjects had unstable conceptual models relating to these tasks; and (c) subjects use a number of modes of representation, either simultaneously or sequentially, as they proceed through problem solutions.

The Subjects

Individual interviews were conducted with 80 subjects, 16 from each grade level, fourth through eighth. There were 46 girls and 34 boys, with approximately equal numbers of boys and girls at each grade level. At each grade, the students were chosen to represent a range of levels of rational-number understanding, as measured by the battery of written tests (described in the third section of this chapter) that were developed by the RN project. Among the 16 youngsters interviewed at each grade, 4 were selected from each quartile.

The Interviews

Each 45-60-minute interview involved the 11 basic problems described below. Some of the problems were presented as "word problems," typed on index cards, and handed to the student. Materials such as papers, pencils, clay pies, Cuisenaire rods, and counters were available within easy reach on the interview table: their use was neither encouraged nor discouraged. The ''concrete problems'' or "real problems'' were presented orally using the materials described in each problem.

The problems were always presented in the order shown below. The entire interview is reproduced here so the reader may refer to the wording and order of the problems.

The Problems

1. The Chocolate-Eggs Addition Problems:

E: Present 6 eggs in a 12-pack carton, and say, ''This is one-half carton of eggs.''

E: Remove the preceding carton, present 4 eggs in a 12-pack carton, and ask, How much is this ?''

E: Repeat the preceding question, using cartons containing: 3 eggs, 8 eggs, 5 eggs, 9 eggs, and 10 eggs.

E: Present 2 eggs in one carton and 3 eggs in a second carton, and ask. How much is this altogether?''

E: Repeat the preceding question, using cartons containing 4 eggs and 5 eggs, respectively.

2. The Chocolate-Eggs Multiplication Problem (Presented Orally):

E: Present one dozen chocolate eggs in a 12-pack carton. Then, pose (orally) the following question, ''Mike took of a carton of eggs to a picnic. He and his friends ate of the eggs. How many eggs did they eat altogether?"

E: Immediately, say, "I'11 repeat the question slowly." (Repeat orally–more than once, if requested.)

3. The Addition Word-Problem:

Jim's family ordered two pizzas for supper, one with sausage and one with mushrooms. Jim ate of the mushroom pizza, and of the sausage pizza. How much pizza did he eat altogether?

4. The Multiplication Word-Problem:

Yesterday, Karen ate of a chocolate cake. Today, she ate of the remaining cake. How much did she eat altogether?

5. The Cuisenaire Rod Addition-Problem:

Present a set of rods, glued onto a posterboard, where a "unit'' rod is created using a x wooden stick, cut 12 cm long.

Then, present a pile of loose rods in the following lengths:

2 12-cm rods to represent the unit, or 1

2 10-cm rods to represent the fraction 5/6

2 9-cm rods to represent the fraction 3/4

2 8-cm rods to represent the fraction 2/3

2 7-cm rods to represent the fraction 7/12

3 6-cm rods to represent the fraction 1/2

2 5-cm rods to represent the fraction 5/12

6 4-cm rods to represent the fraction 1/3

6 3-cm rods to represent the fraction 1/4

10 2-cm rods to represent the fraction 1/6

16 1-cm rods to represent the fraction 1/12

E: Point to the 12-cm rod that is glued onto the posterboard and say, ''This is one unit long.''

E: Then, point in turn to each of the other rods on the posterboard and (in each case) say, ''How long is this?''

E: Give S one of the loose 9-cm rods and say, "How long is this?" Then, repeat the question using the 7-cm rod, and the 6-cm rod.

E: Next, remove the posterboard from the table, give S the 5-cm rod, and say, "How long is this?" Then, repeat the question using the 8-cm rod.

E: Remove the 8-cm rod, put a 6-cm rod and a 4-cm rod end to end on the table in front of S, and say, "How long is this (pointing to the length of the two rods together) altogether?"

E: Remove the preceding rods, put a 9-cm rod and an 8-cm rod end to end on the table in front of S, and say, ''How long is this (pointing to the length of the two rods together) altogether?"

6. The Concrete (Pizza) Addition-Problem:

E: Present precut parts of 6-inch clay ''pizzas" on cardboard ''plates,'' like those in Figure 9.3a, and say "Pretend that these are pieces of pizza.''

E: Present the 1/2 piece and say, ''How much is this?''

E: Present the 1/3 piece and say, ''And, how much is this?''

E: Present the 1/2 piece and the 1/3 piece simultaneously, as in Figure 9.3b, and say, "Now, how much is this altogether?"

E: Remove the preceding pieces, present the 3/4 piece, and say, "How much is this?''

E: Remove the 3/4 piece, present the 5/6 piece, and again ask, ''How much is this?"

7. The ''Realistic'' (Cake) Addition-Problem:

E: Present precut parts of 6 x 8-inch clay ''cakes" on (scored) cardboard ''Plates,'' like those in Figure 9.4, and say, ''Pretend D that these are pieces of cake.''

E: Present the 1/4 piece and say. ''Yesterday, I ate this (pointing) much cake. How much did I eat?''

E: Next, put the 1/4 piece into a covered box (because it was eaten) on the table, slightly to the left side in front of S. Then, slightly to the right side in front of S, present the 1/3 piece, and say, ''Today, I will eat this (pointing) much. How much will I have eaten altogether (pointing to both the ''hidden" 1/4 piece and the 1/3 piece)?''

8. The Area Problems:

E: Show S two red shapes, like the ones shown in Figure 9.5, drawn on a "checkerboard" grid.

Point to the first shape (Figure 9.5a) and ask, "How many squares are in this shape?" Then, point to the second shape (Figure 9.5b) and repeat the question.

E: Show S a red triangle on a sheet of white paper (Figure 9.6a): then cover the shape with a clear acetate grid (Figure 9.6b), point to the covered shape (Figure 9.6c and ask. ''How many squares are in this shape?"

E: Repeat the preceding question, only hand the sheet of acetate to S and use a red trapezoid shape, like the one shown in Figure 9.7.

E. Give S a whole plain Hershey candy bar, saying, "The next problem we are going to do has to do with this candy bar. But, we will use this piece of brown paper to stand for the candy bar. Then, as a reward for answering all of these questions for me, you can keep the candy bar. Is that OK?"

Take away the candy bar, and give S a piece of brown paper, like the appropriate one shown in Figure 9.8. Use the piece of paper that is marked into 10 squares to represent the plain candy bar, or for Problem 10 below, use the unmarked piece to represent the almond candy bar.

E: Say, "Yesterday. Bob's mother gave him a plain Hershey candy bar in his lunch. He ate part of it and saved for today. Today, he ate of the remaining candy. How much of the whole candy bar did he eat altogether?"

E: Immediately, say, "I'll repeat the question slowly.'' (Repeat orally–more than once if requested.)

10. The Almond Hershey Candy-Bar Multiplication-Problem

The almond candy-bar problem was identical to the plain candy-bar problem, except that an unmarked piece of paper was used, and the orally presented statement of the question was:

"Yesterday, Ann’s mother gave her a plain Hershey candy bar in her lunch. She ate part of it and saved for today. Today, she ate of the remaining candy. How much of the whole candy bar did she eat today?" (Note: The order of presentation of Questions 9 and 10 was reversed for 50 of the children in each quartile at each grade level.)

11. The Pencil-and-Paper Multiplication Computation-Problems:

One at a time, present the following two written computation problems. Each problem is written at the top of an otherwise blank sheet of paper.

1/2 X 1/3 =
2/5 X 3/4 =

Results and Discussion

Results of the interviews are presented below. First, the differences in the difficulty of the problems are displayed in a table of success rates for each problem in each grade. Then several of the problems are discussed with respect to (a) solution processes used, in particular, the predominant mode of representation in which a solution was reached; (b) the types of errors that typically occurred; and (c) the difficulty of the item compared with that of related items. The Addition Word-Problem (Problem 3) is discussed in greatest detail; discussion of other problems focuses on results we view as most important, interesting, or unexpected. Many of the more obvious and predictable results are summarized in table form.

The Addition Word-Problem

PROCESSES

All the solution attempts began by translating Problem 3 into written symbols, for example,
1/4 + 1/5 = ___. Seventy percent of the students wrote the correct symbolic expression horizontally; 16% wrote it vertically. The remaining 14% of the subjects made errors in recording, usually omitting one or more of the symbols +, /, or =. It is surprising that, although this problem immediately followed two problems involving concrete materials, none of the students initially used materials or drew a picture to represent the problem.

ERRORS

The number of students at each grade level who gave each response type is shown in Table 9.2. Note that all the students who responded correctly (37% of the total) used an algorithmic least-common-denominator (LCD) method.

DIFFICULTY

Some of the most interesting observations about the Addition Word-Problem stem from a comparison of responses to this item with those to a question posed later (following the structured interview) that asked the subjects to "act out" the problem using clay circular regions to represent the pizzas. Students who had obtained the answer 2/9 for their written calculation of the sum were able to look at the clay pizzas and recognize that 2/9 was incorrect. However, when confronted with their written work on the problem, about half of these subjects maintained that was still correct. The discrepancy between the results obtained from the two different representations apparently did not trouble these children; several explicitly stated something like, "That’s okay! These are pizzas and those are numbers–they aren't the same." Such comments seem to indicate either a belief that mathematical computations (in symbolic form) need not agree with real-world observations (of clay pizzas) or that mathematics is simply unpredictable, so sometimes one obtains one answer and sometimes another for the same problem.

The presence of concrete materials in the follow-up problem did not enhance performance. Whereas 30 children had obtained the correct answer to the word problem (all of them using written symbols), a total of only 20 children arrived at the correct answer when the same problem was posed using concrete materials. Three students attempted to solve the problem using the concrete materials; they all obtained incorrect results. All 20 children who were correct were among the 27 children who persisted in using written symbols to obtain an answer. This means that 7 subjects who had previously obtained the correct answer using written symbols were no longer able to solve the problem using written symbols after it was presented in concrete form. They became confused and reverted to adding both numerators and denominators rather than using the LCD approach as before. This outcome runs counter to the widespread belief that materials make a problem easier to solve because it is more meaningful and real.

The interviewer probed to find whether this difference in performance was the result of rote execution of a meaningless algorithm in the first case. Some of the students reported that, when they wrote + = after reading the word problem, they were thinking about parts of circles, and that when they wrote their solution they again thought about circular regions. For these subjects, the symbols were meaningfully related to stored images of concrete objects–but the images were abandoned in favor of a more powerful symbolic procedure for actually carrying out the computation. The follow-up request that the subjects act out the problem using the concrete materials required a demonstration of the symbolic algorithm relating it to the pieces of clay pizzas. Most of the students recognized their inability to respond to the question using the materials and worked out answers using pencil and paper. We believe it is likely that the LCD procedure, although present, was still an unstable element of the rational-number conceptual models of the 7 students who were no longer able to reach the correct result using written symbols. Attending to the concrete materials somehow contributed to the breakdown of an effective symbolic method for solution.

Although concrete materials often provide a useful representation for some stages in solving a problem, it may be extremely difficult to carry out the entire solution in terms of them. Good students eventually learn to select an appropriate representational system to fit each particular part of a problem situation or each specific stage in the overall solution; this understanding builds slowly and requires coordinating complex within-concept networks and between-concept systems.

The Concrete (Pizza) Addition-Problem

In the Problem 6 sequence, subjects were asked to identify circular clay pieces cut in the sizes 1/2, 1/3, 3/4, and 5/6. The addition question was to find the sum of 1/2 + 1/3 immediately after identifying the two (differently colored) component pieces, which were then put together on a single plate. Table 9.3 shows the number of students at each grade level who responded correctly to each question.

PROCESSES

Identifying 1/2 and 1/3 caused no difficulties for the subjects. Of the other fractions, 3/4 was obviously much easier to recognize than was 5/6. Either the missing piece was identified as , so the piece itself had to be , or the missing piece was used as a measure for cutting the remainder of the pizza. (Note that this procedure would be useful only when the missing piece represents a unit fraction.)

An immediate verbal response (apparently based on perceptual cues, and accompanied by no overt actions) was given by 80% of the children for the 3/4 piece and by 63% of the children for the 5/6 piece. When the interviewer probed ("How did you figure that out?"), nearly all these students did draw lines on the clay or on the plate, but the action typically seemed to be a justification rather than the source of the original response. This was especially apparent for the piece.

Table 9.4 shows the number of students at each grade level using each of the following five types of procedures to name the 5/6 piece: (a) relatively passive perceptual cues were used with no lines drawn overtly; (b) lines were drawn in a seemingly trial-and-error way, apparently to fit some previous perceptual cues; (c) the missing piece was used as a standard for cutting the remaining pizza, with the hope that cuts of this size would divide the remaining pizza into a whole number of equal parts; (d) the remaining pizza was cut into equal-size pieces, presumably with the hope that cuts of this size would also fit the missing piece; and (e) the plate was divided into equal-sized pieces (as though the missing piece had not yet been removed), with the hope that cuts of this size would simultaneously fit both the remaining pizza and the missing piece.

Overall, only 25% of the students correctly identified the 5/6 piece. Among these 20 students, 15 used a procedure that appeared to be based on the missing piece, 3 seemed to focus on the pizza ( i. e., their first cuts were not at all the same size as the missing piece), and only 2 successfully used trial and error guided by some sort of perceptual cues.

Table 9.5 shows the number of students at each grade level using various solution procedures for the concrete addition-problem ( 1/2 + 1/3). Twelve of the 14 successful subjects used a written symbolic procedure; only 2 students obtained a correct answer using a concrete procedure.

Because the concrete addition-problem is more complex than the identification of single pieces, it was more difficult to sort solution procedures into distinct categories. Many students went back and forth among (or combined parts of) several of the six basic procedures shown in Table 9.5. For example, some of the students who were classified as using a written symbolic procedure actually began by trying to find an answer using a concrete procedure. Interestingly, however, no student who began by using a written symbolic procedure later switched to one of the concrete procedures.

Some interesting differences appear in a comparison of Tables 9.4 and 9.5. Even though the addition problem was presented in concrete form, 51% of the subjects used paper-and-pencil procedures to solve it. The presence of the two differently colored pieces for the addition problem seemed to draw subjects' attention to the pizza that was present, so more solution attempts were based on the focus-on-pizza procedure rather than the missing-piece procedure, which produced such good results for the single piece.

ERRORS

Among the errors that occurred in relation to concrete procedures for the identification of the 5/6 piece, 63% involved too many cuts, 37% too few cuts. In nearly all cases, errors were related to the fact that pieces were not all cut the same size (nor the same size as the missing piece). The fraction name given as an answer often did not correspond to the visual representation. Some students counted n pieces present and gave 1/n as an answer. Others counted m spaces in the missing piece and n pieces present and gave the answer m/n (instead of n+(m + n)), indicating some confusion between part-whole (fraction) and part-part (ratio) relationships.

For the concrete addition-problem 1/2 + 1/3, the most frequent error committed by subjects using a concrete procedure was to divide each of the two pieces in either halves or thirds and conclude that the sum was 4/5 or 6/7.

DIFFICULTY

For the concrete addition-problem 1/2 + 1/3, the overall success rate was 17.5% (14 of the 80 students), compared with 25% for the identification of the 5/6 concrete piece and 37.5% for the addition word-problem ( 1/4 + 1/5, discussed above). Because the concrete problem was presented after the word problem in the interview, and and are generally better understood than and respectively, the greater difficulty with the concrete problem is somewhat surprising. It is therefore interesting to trace the procedures and the accuracy of the 80 students on the word problem and then on the concrete problem (see Figure 9.9).

Out of the 33 students who used concrete materials to solve the concrete addition problem only 2 were successful. On the other hand, 12 of the 40 who used written symbols on the concrete problem were correct. Eleven of these 12 had previously obtained the correct answer using written symbols on the addition word-problem. Notice that switching from a written symbolic representation to a concrete representation that matched the mode in which the problem was presented proved to be a poor strategy, only 2 out of the 12 who switched were correct, compared to 11 correct out of the 18 who persisted in using written symbols.

Still, pencil-and-paper solution procedures appear to have been somewhat more difficult when they were applied to the ''concrete'' situation than to information given in the ''word" problem. This is consistent with the results of the follow-up questions to the "word'' problem in which giving the students additional concrete aids actually made the ''word'' problem more difficult for some students. Perhaps, for some students, even when a problem is immediately converted to a written expression such as 1/2 + 1/3 = ___, what is going on in the student's mind may be slightly different for a "word" problem than for a "concrete" problem.

The "Realistic" (Cake) Addition-Problem

A problem that involves concrete materials, like the one in the preceding section, is not necessarily real in the sense that it would be likely to occur in an everyday situation. It is unlikely that someone would actually put two pieces of pizza together and ask, "How much altogether?" Even if the question were asked, reasonable answers would vary from saying, "That much," pointing to the pizza, to "Almost a whole pizza''; it would be surprising to see someone take out pencil and paper to calculate a response. In real situations, quantitative information usually is given for some purpose; information is stated or recorded at a level of precision that is reasonable in terms of both the source of the information and the use to which it will be put. Estimation, approximation, and rounding off reflect important properties of the models we use to describe real situations. Unfortunately, successful performance on textbook word-problems often requires the suspension of everyday criteria for evaluating the sensibility of realistic questions and responses, with students learning, instead, to give answers that teachers and textbooks expect, sensible or not.

Another characteristic that distinguishes many real problems from their word problem counterparts is that, in real problem-situations, the relevant information is not necessarily all given in the same representational mode. For example, in real addition-situations that involve fractions, the two (or more) items to be added may not occur as two written symbols, two spoken words, or two cakes; the addends may be one piece of cake and one written symbol, one fraction word and one written symbol, or (in our ' realistic ' addition-problem) one fraction word and one piece of cake. This is because, when symbols (written, spoken, or concrete-pictorial) are used to represent something, it is often because the thing being represented is not present spatially or temporally. Thus, part of the difficulty for students in these multiple-mode situations is to translate both addends (as well as the answer) into a single representational system. Our ''realistic" problem exemplified this type of problem, virtually ignored in textbooks, which inherently involves more than one representational system. In the problem the subject first identifies 1/4 of a cake, which is presumably eaten, thus hidden, then the subject is shown another 1/3 of the cake to be eaten, and is finally asked how much cake will have been eaten altogether.

PROCESSES

Table 9.6 shows the number of students at each grade who used each of four basic response types for the ''realistic'' addition problem. None of the concrete solution attempts was successful. Forty-three students (54%) used pencil-and-paper procedures, which was slightly higher than for the concrete problem. Drawing pictures and using gestures (accounting for 31% of the responses), as well as many of the responses in the miscellaneous category, were never used on either the concrete addition-problem or the addition word-problem discussed above.

In this problem, because the 1/4 piece was hidden, some representation of the hidden piece was required. The type of representation chosen (i.e., picture, gesture, symbol) directly influenced the solution procedures and errors that were made.

ERRORS

Students who drew a picture of the hidden piece often (in 8 out of 16 cases) drew a picture of the piece that was visible. Apparently, they were uncomfortable having one of the addends as a picture while the other was a ''real'' piece of cake (i.e., having addends in two different representational systems). The most common (incorrect) responses were: (a) 2/7: the shaded pieces (see Figure 9.10) were counted, and then the pieces (again, see Figure 9.10) were counted: (b) 2/5: the shaded pieces were counted, and then the unshaded pieces were counted. (Note: This latter response points again to the confusion, quite common in children's primitive rational number thinking, between the part-whole relationships that are relevant for fraction situations and the part-part relationships between distinct quantities that are relevant for ratio situations.)

Students who used a gesture (usually accompanied by verbal descriptions) to represent the hidden piece, often used their hand to indicate how much more of the plate would be covered by the hidden piece. Figure 9.11 illustrates roughly the procedure that was used. The most common (incorrect) response was ; that is, the size of the hidden piece was distorted significantly in order to fit an answer that the students considered ''nice.'' (Note: The predominance of in the rational-number thinking of children is well known. For example, see Kieren & Southwell, 1979.)

Students who used a written symbol to represent the hidden piece made errors and gave answers similar to those discussed in connection with the concrete problem. There was again evidence that some students shifted from one representation to another, and from internal to external representations, during the solution process.

DIFFICULTY

None of the subjects who used concrete procedures obtained a correct result. Of the 43 children who used written symbols, 19 were correct, which represents 24% of the total. There was a higher success rate among subjects using written symbols for this problem than for either the concrete addition or the addition word-problems discussed above. Perhaps adding
1/4 + 1/3 was easier because it is one of the most popular demonstration problems appearing in textbooks to illustrate the LCD method for adding fractions, and so was familiar to the students.

The Chocolate-Eggs Addition-Problem

Given a carton of 12 eggs as a unit, subjects were presented various numbers of eggs and asked to identify the fraction represented by each (1/3, 1/4, 2/3, 5/12, 3/4, 5/6). They were then asked to find two sums (1/6 + 1/4 and 1/3 + 5/12) for which the two addends were displayed as sets of eggs in two separate cartons.

PROCESSES

Virtually none of the students used paper and pencil for the egg addition problems. Characteristics of the materials themselves apparently facilitated the higher rate of correct responses (see Table 9.1). First, the carton itself was always present as a frame to remind subjects of the whole unit. Focusing on the carton as the whole and the eggs as the parts, with distinct vocabulary to refer to them, made it easier for subjects to keep both in mind while making judgments about part-whole relationships. The fact that the whole consists of discrete objects rather than a continuous quantity (such as an area or length) made it easier to subdivide into unit fractions consisting of equal-sized sets of eggs. These characteristics of the materials also made part-part relationships easier to notice, which caused difficulties for some subjects.

ERRORS

Compared with errors occurring on the other concrete addition problems, part-part errors were more common on the egg problems. For example, for naming 5/12 (the easiest identification problem), the most frequent incorrect answer was 5/7. When the subjects' attention was focused on the part, that is, the five eggs in the carton, they seemed to lose track of the whole, responding with the ratio of eggs to empty spaces. This was, therefore, another situation in which part-part ratio ideas were sometimes confused with part-whole fraction ideas.

Another source of confusion was the relationship between the number of objects and the sizes of unit fractions. For example, 1/3 was more difficult to identify than 5/12 because four eggs had to be recognized as one-third. Similarly, 2/3 was still more difficult, because eight eggs had to be recognized as two-thirds. In each case the whole carton had to be partitioned into unit fractions consisting of equal sets of objects, then some number of these sets had to fit the part in the part-whole relationship.

For the egg addition problems, the presentation of the two addends in separate cartons was a source of confusion that was not present in other questions. Thus, 9/24 was one popular incorrect answer for 1/3 + 5/12. Another common incorrect response was 9/15, the ratio of eggs to empty spaces in the two cartons. In both error types the students lost track of the unit.

DIFFICULTY

Table 9.1 shows that the chocolate-egg problems were considerably easier than problems involving the same fractions but different materials or modes of representation. In spite of the increase in the salience of part-part notions, the characteristics of these materials, described above, are apparently responsible for the subjects' greater success.

The Cuisenaire Rods Addition-Problems

For the Cuisenaire rods addition-problems, like the egg problems, the parts and wholes were separate and easy to identify, and appropriate unit fractions were fairly easy to recognize; these characteristics, in addition to the subjects' familiarity with the rods, contributed to relatively good performance (see Table 9.1). On the other hand, rods are continuous rather than composed of discrete pieces, and there is no ever-present frame to maintain the size of the unit as there is for the egg problems.

The typical solution procedure for the Cuisenaire rod problems consisted of three steps. The first step was to find an appropriate unit fraction piece that fit a whole number of times into both the 12-cm unit rod (call it n) and the number of pieces that fit the length being measured (call it m) were counted. Finally, if these steps were completed correctly, the answer was m/n.

Thus, unlike the case with the egg problems, for which the answers could almost be read from the materials, the Cuisenaire rod problems required the chaining of several (relatively simple) steps. The part-whole judgments needed to find an acceptable unit fraction piece depend on within-concept networks in the subject's rational-number conceptual model; the measuring and labeling of lengths of the part and the whole depend on between-concept systems (relating measurement and fraction ideas) and representational systems (to accommodate parallel processing of the concrete objects as visual stimuli while attaching spoken symbolic labels during counting).

The Area Problems

For the area problems. the goal was to find how many square unites there were in a nonrectangular figure; specific fractions could not be abstracted and plugged into a written algorithmic procedure. These problems were very much embedded in the graphical pictorial representation that was given to the subjects.

Nearly all the students used the following procedure on the area problems: First, they counted all the whole squares. Then, they looked for parts of other squares that would fit together to make wholes.

Finding pieces to make wholes seemed to be a relatively simple task for most students. When errors were made on the area problems, they usually resulted from memory overload. That is, the student would forget how many whole pieces had been counted when he or she went on to find pieces to make up more wholes, or would lose track of which pieces had already been used in making previous wholes. Throughout this counting process, it was striking that virtually none of the students used any recording system either to count or to keep tack of the counted pieces. They relied entirely on internal memory. Still, their errors tended to be within +1 or -1 of the correct answer.

The Multiplication Problems

Six different multiplication problems were given. One problem was a word problem whose solution can be characterized by the expression 1/4 + 1/2 X (1 - 1/4). Two of the problems involved Hershey candy bars, one using a plain candy bar whereas the other used a candy bar with nuts; this was an important distinction because the plain bar is scored into 10 squares. The fourth problem was also a concrete problem, involving chocolate eggs in a 12-pack carton. The last two problems were straightforward computation problems, 1/2 X 1/3 and 2/5 X 3/4.

The success rates for the six problems were included in Table 9.1. The computation problems were the easiest, then the problem using the eggs, the candy bar with nuts, the word problem, and the plain candy bar.

The Multiplication Computation Problems

The multiplication computation was quite easy, although probably for the wrong reasons. Recall that the most common (incorrect) computation procedure for addition involved adding the numerators and denominators. For multiplication, a comparable procedure yields a correct answer.

The Multiplication Word-Problem

PROCESSES

For the multiplication word-problem, which was slightly more difficult than the earlier addition word-problem, the solution procedures were entirely different. For the addition problem, no one drew pictures, and the only successful procedures involved pencil-and-paper computations. For the multiplication problem, the attempted solution procedures are shown in Table 9.7. Sixty-five percent of the students used spoken language and/or drew pictures; almost half of these were successful. None of the students using exclusively pencil-and-paper procedures was successful.

ERRORS

On the word problem, if a picture was used, errors occurred most often when subjects drew two pictures, one showing and another showing . The students did not know what to do with the two separate pictures, and often resorted to doing "some number thing'' with some of the numbers 1, 3, 4, 1/3, or 1/4. Similar errors were committed by students who initially translated the word problem into written symbols. The fractions most frequently written were 1/4 and 1/3, not (for example) 3/4, the amount remaining after the first piece of cake was removed.

Concrete Multiplication Problems

Looking at success rates in Table 9.1, one of the most interesting results is the radical difference in difficulty among the three concrete problems, two of which were closely related. The explanation for this fact has been discussed in connection with several of the addition problems. Slight differences in materials often greatly hinder or facilitate a student's ability to use a system of rational number relations and operations. Having a conceptual model and being able to use it in a given situation are quite different. A student's ability to use a given conceptual model depends considerably on the stability (i.e., degree of coordination) of the constituent structures. For many of our fourth- through eighth-grade subjects, rational-number concepts were apparently relatively unstable, particularly with respect to the task of translating from one representational system to another.

PROCESSES AND DIFFICULTY

Tables 9.8., 9.9, and 9.10 display the number of subjects who arrived at correct and incorrect answers using each solution procedure. The use of the concrete materials that were part of the problem presentation was least helpful for the plain candy-bar problem in which the division of the rectangle into 10 squares probably interfered with attempts to divide the bar into thirds. The concrete materials were most facilitative for the chocolate-eggs problem in which the materials lend themselves to a ready representation of the fractions in the problem; that is, 1/4 of 12 eggs is 4 eggs, 3/4 is 9 eggs, 2/3 of 9 eggs is 6 eggs, or 1/2 of the whole carton. For the candy-bar problem with nuts, the unmarked rectangle neither contributed to nor interfered with the subject's ability to impose fourths.

Further, the plain candy-bar problem was the one for which a language representation would be least helpful. Finding 2/3 of 1/2 can be quite a different problem from 1/2of 2/3, the commutative property notwithstanding. For 1/2 of 2/3, subjects can find an answer thinking of the problem as analogous to 1/2 of two objects, for which the result is one object, or, in this case, 1/3. This type of language representation, implemented either overtly or covertly, would facilitate reaching a correct result in the candy-bar (with nuts) problem ( 1/3of 3/4) and the chocolate-egg multiplication problem ( 2/3 of 3/4). It was not helpful for the plain candy-bar problem ( 2/3 of 1/2).

Apart from comparisons among the concrete multiplication problems, it is of interest to compare solution procedures on these concrete problems with those on the concrete addition-problems discussed above. The most striking difference is the drastically reduced rate of written symbolic procedures on the multiplication problems, accompanied by a much greater reliance on concrete procedures.

Language representations, useful for some of the multiplication problems, did not occur on the addition problems.

The data collected in the Rational Number Task Interviews support the following generalizations, some of which are based on the data as a whole, and some of which stem from results on one or more particular items. Reference is made to the problem or problems that were especially relevant.

1. Realistic word-problems often differ significantly from isomorphic concrete or real-world problems in difficulty, preferred solution processes, and the types of errors most commonly made. In fact, virtually any small change in the task had a noticeable impact on performance for most of our fourth- through eighth-grade subjects. This variability is regarded as an indication of the instability of subjects' rational-number conceptual models.
   1. Problems presented in terms of concrete materials are not necessarily easier than those presented orally or using written language and symbols. For a number of children, performance declined when they were encouraged to use concrete aids to help in solving problems. Compare, for example, the results on the addition word-problem with its concrete follow-up question and with the concrete addition-problem. Some subjects were able to execute the written LCD procedure correctly on the word problem but were unable to do so (even when they again were working in written symbols) after concrete materials had been brought into the problem situation.
   2. The representational system favored by subjects was distinctly problem specific. For example, subjects were much more inclined to translate the addition word-problem into written symbols than the multiplication word-problem. Even within the class of concrete problems, we found, for example, that the chocolate eggs were used for finding answers whereas the circular regions were not. The fact that the specific fractions involved in the problems may have influenced the results only strengthens the point.
   3. Some successful solution procedures are closely tied to specific problem-representations and are not consistently "called up" for solving an isomorphic problem presented in a different representational mode. For example, the missing piece strategy used by subjects to identify the 5/6 circular region was apparently unavailable for finding the sum of 1/2 + 1/3 even when circular pieces were placed next to each other on the same plate.
   4. There was a great deal of evidence of confusion between part-whole fraction ideas and part-part ratio ideas; to whatever extent this was present in the subjects irrespective of the stimulus, it appeared that certain materials (e.g., the chocolate eggs) elicited more inappropriate part-part responses than other materials.
   5. The mode of representation was important not only in the initial problem presentation, but also when selected by the subject. For example, on the "realistic'' addition-problem, the mode in which the subject chose to represent the hidden 1/4 piece seemed to affect the course of the solution dramatically. On particular multiplication problems, for which the denominator of the first factor was a divisor of the numerator of the second factor, a language-based representation apparently facilitated correct responses.
2. Representation of a problem situation is a very active process.
   1. During the course of a solution attempt, subjects often changed their external representation of the problem from one mode to another. For example, very few subjects solved the concrete addition-problem in the concrete mode; most changed to written symbols.
   2. Subjects often used two or more systems of representation simultaneously. The most obvious occurrences were in response to the multiplication word-problem for which a large number of subjects used spoken language and pictures, and the Cuisenaire rod addition-problems, for which many subjects monitored their solution steps by narrating their actions.
   3. Internal and external representations of the problem situation interacted with each other and influenced the course of the solution. On several problems (for example, in identifying concrete, pizza circular regions) subjects responded based on perceptual cues, then constructed post hoc explanations of their answers for the interviewer. For the "realistic'' addition-problem, the strength of the understanding of the fraction 1/2 led many children to adjust their external representation of 1/4 + 1/3 so that would look like an appropriate response.
3. Some beliefs children have about mathematics and about how to respond to mathematics problems affect performance.
   1. Most of our subjects seemed to be working toward precise answers as the goal of solving problems; there were few instances of estimates or approximations. This tendency was most apparent in response to the more typical textbook word-problems and least apparent on the area problems.
   2. Several children were unconcerned about obtaining different results for the addition word-problem when they used concrete materials versus paper-and-pencil procedures

Rational-Number Written-Test Results

Written language, written symbols, and several types of pictorial representations were used in assessing children's rational-number understandings in the set of written tests developed in the RN project. These tests, the subject of this section, provided the baseline data about paper-and-pencil modes of representations and fraction and ratio within-concept networks that the interviews described above were designed to extend and explore.

Three paper-and-pencil tests were developed for the RN project: the Assessment of Rational Number Concepts (CA), the Assessment of Rational Number Relationships (RA), and the Assessment of Rational Number Operations (OA). The first tests basic fraction and ratio concepts. The second tests understanding of relationships between rational numbers, involving ordering, equivalent rational forms, and simple proportions. The third tests abilities to perform addition and multiplication operations with fractions. This section of the chapter identifies the rational-number characteristics tested and summarizes some general results from two of the tests (the OA and the CA). Further information on the Rational Number testing program appears in Lesh and Hamilton (1981).

The written tests were administered to about 1000 students in Grades 2 through 8 in Evanston, Minneapolis, DeKalb, and Pittsburgh between early November and late January of the 1980-1981 school year. The tests were prepared in two parallel versions, with most of the students taking Version I. The tests were administered in a modularized form so that a core of items was given to all grades, younger students did fewer items than older students, and the most difficult items were done only by older students.

Two of the tests, the CA and the OA, are referred to extensively in this section. Each item from those tests is reproduced in Appendix 9.B, with data on the fourth through eighth graders who took the texts. In all, 650 fourth through eighth graders took the CA and 608 took the OA. The items for each test are arranged in order of percentage correct for the students who were given the item. The purpose of this arrangement is to enable a broad overview of order of difficulty for the items on each instrument. A sample item from the Concepts test is given in Figure 9.12.

Beneath the item are two lines of information, which can be interpreted as follows: "Item C-8" refers to the overall rank (by proportion correct) of this item on this test. Thus, out of the 60 items on the Concepts test, this one had the eighth highest score. Next, "(5)" refers to the original number of the item on the test (and corresponds to the number next to the item's question stem). "Gr. 4-8, n = 650" means that Grades 4-8 were given the item, involving 650 students. Next is a set of letters and numbers corresponding to the answer choices: "a)2 b)596 . . . " This means that 2 students out of the 650 selected choice a, 596 chose b, and so forth, "na/1" means that one student gave no answer.

On the second line, two results for each grade that took the item are given: the proportion correct for that grade, and the item rank for that grade. Thus. "G4-.797 (7/43)" means that 79.7% of the fourth graders did this item correctly, and this item was the seventh easiest out of 43 items for the fourth graders. Item ranks varied across grades on the items. Notice, for example, that this item was the sixth easiest (out of 60 items) for the sixth and seventh graders, but the fourth easiest (out of the same 60 items) for the eighth graders.

Two caveats are in order. First, because the ordering of these items collapses across all grade levels, the order of difficulty of an item is somewhat different from its order for all grades that did that item. Second, all items are included in the list, even though not all grades did all items. Thus, for example, the rank order of 7 on item 8 for the fifth graders is out of 43 items, whereas the rank of 6 for the sixth graders is out of 60 items (which include all 43 items done by the fifth graders).

Items on each test are identified along several dimensions, including type of representational translation, rational-number size, and rational subconstruct (fraction or ratio). A list of characteristics for each test is included in Appendix 9.B, along with a generating scheme for the test items.

Technical Information

Cronbach-alpha reliabilities across the tests averaged 0.881 for all the tests, excluding the short 15-item OA given to fourth graders, which had an alpha reliability of 0.489. Within-subject reliabilities averaged .850, excluding the fourth grade OA, which had a within-subject reliability of 0.318. One external validity measure involved the correlation between eighth-grade performance on six OA items and six corresponding National Assessment items administered to 13-year-olds in 1979. Although item stems in each pair were similar or identical, answer sets were different in that the OA limited responses to five given choices or omit, whereas the National Assessment items required students to write in their answers. Students scored higher on all OA items, given answer choices, than their National Assessment counterparts on comparable items with no answer choices given. The high correlation between the six score pairs (0.918), and the fact that one would expect an incrementally higher score for items that give answer choices compared with those that do not, are evidence of the comparability of the eighth-grade sample with the National Assessment sample.

Discussion of CA and OA Results

A plethora of observations from these results are readily made. This section will identify and discuss a few of these, including:

1. A difficulty ordering of translations between representational modes.
2. The easiest and most difficult problems from each test.
3. Notable jumps in performance from one grade to a later grade, including examples of apparent declines from one grade to another.
4. Item pairs that suggest the "nonadditivity" of some rational-number understandings.

Ordering of Translations between Representational Modes

Representational translations on the CA were

1. Symbols to written language (e.g., C-3).
2. Written language to symbols.
3. Picture to pictures (e.g., C-12).
4. Written language to pictures (e.g., C-17).
5. Picture to written language (e.g., C-20).
6. Symbols to pictures (e.g., C-32).
7. Picture to symbols (e.g., C- I 1).

Because each item on the test can be characterized by one of these seven translations, one can think of the CA as consisting of seven "subtests." Considering the 43 items that were done by all five grades, the order in which the translations are listed above proved to be the order of their increasing difficulty for each grade, with only minor exceptions. The fourth and fifth graders found the written-to-symbol subtest more difficult than the picture-to-picture translations and more difficult than the written-language-to-picture translations. Also, the seventh graders did slightly better on the picture-to-written translations than on written-to-picture translations. Table 9.11 shows the representation translation success rates for each grade.

Such an ordering is plausible. The easiest translations are those that involve simply reading a rational number in two different modes, requiring little or no conceptual processing of the meaning of the rational number (e.g., C-4). The relative lack of familiarity with the symbol notation provides a reasonable explanation for the apparent difficulty the early fourth and fifth graders experienced with the items involving four symbolically expressed rationals in the answer set (e.g., the written-language-to-symbol subtest). Next are those that require mapping one picture to another that is isomorphic with respect to the fraction shaded (e.g., C-24). Following that are translations between pictures and written language, followed by translations between pictures and symbols. The written language representations of rationals appear to be easier to process than the symbolic coding of rationals. As will be discussed later, this may be attributed to the fact that symbolic representations do not encode rational components (i.e., numerator and denominator) differently, though they have different meanings, whereas written language representations do express each component in a different form. For both translation types, those involving written language and those involving symbols, the translations to pictures were more difficult than from pictures. The former translations include four pictures in the item answer set, whereas for the latter, only one picture appears in the item stem. Translations to pictures thus demanded more visual processing than did translations from a single picture in the item stem. Thus, the data support the implication that a written or symbolic expression is easier to process than is a pictorial representation.

Easiest and Most Difficult Items

The single exception to the assertion that "a written or symbolic expression (involving no conceptual processing) is easier to express than a pictorial representation" is in item C-1. This item asked students to identify 1/2 as the shaded fraction of a circle. Earlier research has affirmed the primacy of "halfness" in children's earliest rational-number understandings (e.g., Kieren. 1976). This was also the case on the OA. Item 01, involving "giving away half of six puppies," was the easiest item for all grades except sixth, for which it was second easiest. Otherwise, the easiest items on OA involved the interpretation of concrete situations with simple part-whole ideas, expressed in written language.

The most difficult items on both tests required what could be called second order processing of the part-whole idea. For all grades, C-60 was the most difficult CA item. It required interpreting each third of the configuration of nine circles as a whole, as per the given information. Understandably, the most popular distracter was 7/9, selected by 36% of the students. Furthermore, this item elicited many more part-part responses ( . selected by 16% of the students) than other items involving discrete objects (such as C-5, for which less than 7% of students selected either of two possible part-part distracters in the answer set).

A similar item was the fourth most difficult item on the CA, given only to sixth through eighth graders. C-57 required students to interpret each large rectangle as a third, with three large rectangles thus comprising a whole. For this item, nearly 40% of the students selected the response interpreting each large rectangle as a whole, rather than as a third partitioned into fourths (or twelfths) of the whole.

''Fractions involving fractions'' also proved to be among the most difficult on the OA. Less than 1 in 6 seventh graders and barely 1 in 5 eighth graders interpreted 0-32 as involving a fraction of a fraction requiring multiplication. Even fewer students correctly interpreted a similar item, 0-34. Barely 1 in 10 seventh graders and 1 in 6 eighth graders could tell how many thirds equal 1/5; more than half of the seventh and eighth graders simply answered "cannot be done" (0-33). The prevalence of this response is ironic because rational numbers are the first number system children encounter that is closed with respect to all four basic operations. The most difficult item on the OA was 0-35, requiring the student to process a fourth of a half and a half of a half and then find their sum. Fewer than 1 in 12 seventh and eighth graders selected the correct response for this item. For this particular item, a visual estimate of the fraction shaded in each half of the picture was required, and this surely contributed to its difficulty. Interestingly, however, immediately preceding this item on the original test was 0-14, for which the student also had to compute a fourth of a half, and a half of a half, and then their sum. On this item, however, we showed the whole rectangle partitioned, and explained each of the possible responses. Nearly one-half selected the correct response on that item, though hardly any could do the next problem, 0-35, which was very similar and which had the same answer.

Jumps from One Grade to Another

The three largest performance jumps from one grade to another occur between the same two grades, fifth and sixth, for items 0-5, 0-6. and 0-12. The average increase from Grade 5 to Grade 6 on these items was 43.2%, compared with an average increase of only 11.2%. All three of the largest jumps involved addition or subtraction of fractions with like denominators. One can infer a significant instructional effect between early fifth grade and early sixth grade with respect to this skill. These items provide an interesting contrast with 0-7, another fraction addition-problem with the same denominator. Fifth graders performed significantly better on this item, for which the denominator was expressed in written language rather than in symbols. The most popular distracter for 1/3 + 1/3 was .2/6 Understanding the numerator of each addend as a cardinal value is familiar to the students. The denominators, however, look like standard cardinal numbers, but their meaning is quite different. Problems such as 0-5, with representations that preclude attaching the same meaning to numerator and denominator, do not require counterintuitive processing or algorithms for standard fraction symbols. Attaching different meanings or algorithms to numbers with the same form, but different position in the fraction, is the principal instructional achievement discerned by the OA, and it occurs between early fifth and early sixth grade.

Jumps ''backward" from one grade to the next also merit comment. On the CA, sixth graders outperformed seventh graders on four out of five consecutive items on the original test (52-55, and C-7, C-56, C-34, and C-37 on the reordered version in Appendix 9.A). Three items required converting representations from mixed to improper form, and vice versa.

The greatest proportion of "downward jumps" occurred between sixth and seventh grade. On the CA, seventh graders as a whole averaged 71.3%, versus 67.8% for the sixth grade. On the OA, sixth graders actually scored slightly better as a group than did seventh graders, by a margin of 50.8 to 49.9% (considering only the 28 items done by both grades). This suggests the possibility that fraction operation skills from Grade 6 to Grade 7 are unstable and that instruction barely maintains or only slightly improves their level.

Task Variable Additivity

A natural question would be the feasibility of devising a hierarchy of rational number task variables that would enable prediction of performance on rational number tests. Such a hierarchy would involve variables v1. . . vn, with additive properties such as

(v1 + v3) - (v1 + v4) = d ---> (v2 + v3) - (v2 + v4) = d

or

(v1 + v2) > (v1 + v3) ---> (v4 + v2) > (v4 + v3),

where the sum (vi + vj) represents the expected performance level for a rational number task comprising two task variables, vi and vj, where the difference, d, between two tasks is the expected performance difference for a student population, and where order, >, is defined by order of difficulty among tasks.

Several examples from the CA and the OA suggest that such a hierarchy is not plausible. They indicate that the impact of changing a single variable in a pair of items may either be much greater than or possibly even opposite to a similar or identical variable change for another pair of items.

EXAMPLE 1

The task variable change is from multiplying 2 fractions of the form p/q (v1), to multiplying a whole number by a fraction (v2). Sample item pairs are (0-18, 0-8) and (0-19,0-34).

Students found whole-number-fraction multiplication significantly more difficult than fraction-fraction multiplication with a symbol-only representation, such as Items 0-18 and 0-8. However, in the context of a brief word-problem, the same variable change produced the opposite result, with students finding the fraction-fraction more difficult than the whole-number-fraction multiplication. In this case, if v3 is symbol mode computation, and v4 is word-problem context, we would have:

O-18 O-8

(v3 + v2) > (v3 + v1) but

(v4 + v2) < (v4 + v1)

O- 19 O-34

whereas, under the assumption of additivity, if v2 > v1, and if (v3 + v2) > (v3 + v1), it would follow that (v4 + v2) would be greater than (v4 + v1).

EXAMPLE 2

The task variable change is from continuous pictorial representation (v1), to discrete object representation (v2). Sample item pairs are (C-39, C-60) and (C- 17, C-8). Students found item C- 17, with a continuous representation, more difficult than the discrete item representation in C-8. C-39 contains a continuous representation of the whole, and proved to be one of the easier items on the CA. In contrast, C- 17, representing a whole with three discrete objects, was the most difficult item on the CA. The intrinsic nature of a discrete object representation of the whole in C-60 elicited a special distracter, the one that interpreted the entire configuration as a whole. For these two item-pairs, if v3 is written-to-picture identification of a simple fraction, and v4 is, given a whole, identify a fraction greater than 1, we have

C-17 C-8

(v1 + v3) < (v2 + v3)

but

(v1 + v4) > (v2 + v4)

C-39 C-60

Conceptual-Model Processes in the Written Tests

Although applied problems evoke richer transformations in conceptual models than those observed on the CA and OA, it is helpful to consider such processes on those instruments. Such a discussion, pertaining to any particular test item, would involve three components and the interactions among them: the conceptual model brought to bear on the problem, the item stem, and the item answer choice set.

The ''interactions'' depicted in Figure 9.13 involve such processes as imposing meaning in the direction of the arrow. For example, a conceptual model imposes a meaning on the representation "1/2".

The conceptual model directs two different process types when the student is doing these kinds of paper-and-pencil items. The first is (within-stem) or (within answer-set) processing, for which rational-number meanings are imposed or attached to each representation in the stem or answer set. The second processing type involves the translations between the stem and the answer set (that contribute to the meaningfulness of within-stem or within-answer-set processing), and which compares meanings imposed on the stem with meanings imposed on the answer set until a "fit" is achieved. The conceptual model effectively encodes and processes both the stem and the answer set and imposes a structure on each. Translations back and forth between the stem and the answer set continue until an isomorphism is established between the stem and one choice in the answer set. Every time the model translates the stem structure to the answer choice set and fails to make an isomorphism, it reprocesses the representations in each and perhaps modifies the meaning or interpretation attached to each representation. In this sense, the translation contributes to the within-stem and within-answer-set processing.

The role of translations, vis-à-vis the amount of processing required to read symbol or written-language representations, merits comment. First, the easiest translations were those that required no meaningful rational number conceptual understandings (e.g., C-3). The data suggest that youngsters can do this without part-whole understandings. It may be that written or symbolic expressions, such as those in C-3, are intuitive and unstable for some children, as are some of their pictorial understandings of part-whole relationships. It may be, however, that when a symbol, for which a student has an unstable or intuitive model, appears with four pictures that illustrate part-whole understandings that the student can identify only intuitively, the understandings in the two modes intersect and the student selects the correct response. The translation stabilizes the understandings associated with the two representations by implicitly saying to the student "whatever is in Mode A means the same thing that is in Mode B, so the overlap of understandings you have between the modes is the correct understanding within each mode." In this way, the multiple-choice items serve to refine or stabilize a model.

An example in which this may occur is Item C-5, which was intended to distract students with part-part understandings. It appears to have had the opposite effect. Students who did not come to the item with models differentiating part-part from part-whole had the opportunity for the item answer-set to facilitate such a differentiation, by giving two examples of part-part (c and d) and only one example of part-whole (b), with the explicit assumption that only one answer is correct. Students could therefore deduce the correct response and learn something in the process.

Thus, it is possible to view each response as a function of six variables:

1. Within-item-stem processing, directed by the conceptual model.
2. Within-answer-set processing, directed by the conceptual model.
3. Translations and matching, from item stem to answer set.
4. Interaction between conceptual model stability and available distracters (i.e., in the answer set).
5. Modification of the conceptual model by previous items or the current problem.
6. Error term.

Concluding Remarks

Research on the acquisition and use of mathematical concepts and processes is influenced by the researchers' theoretical perspectives, whether or not they are explicitly stated. The investigators' beliefs about how cognitive structures are organized and interrelated in the minds of their subjects shape the questions that are chosen to be addressed, the methods and tasks that are selected, and what is regarded as important among the data that result. The chief goal of this chapter has been to define and illustrate our present understanding of a theoretical construct, the conceptual model, that underlies our program of research on applied mathematical problem-solving. The rational-number data presented here, collected from written tests and structured interviews, are highly relevant to our applied problem-solving research because the rational-number system is one of the most sophisticated systems familiar to middle-school youngsters and because the variety of subconstructs provides a rich context for applications appropriate to our subjects.

We have focused on within-concept networks and systems of representations, the two components of conceptual models that are most elementary and crucial to an understanding of the construct. These components have been described in relation to rational-number conceptual models both because they are more discernible in that context than they would be if we tried to describe them in more complex applied problem-solving situations, and because it is apparent how our interest in the growth and use of these components provided a framework and direction for our research. The other two components of conceptual models, between-concept systems and dynamic (modeling) mechanisms, are addressed in our current applied problem-solving research.

We have described three approaches to the construction of tasks for research focusing on the growth and development of conceptual models used by middle school students to solve realistic problems involving mathematical ideas. Tasks may originate in the mathematical ideas themselves, or they may be created to be isomorphic to typical textbook word-problems, or they may be derived from real problem-situations in which the use of mathematics arises naturally. The research included in this chapter has taken the second of these approaches. Furthermore, it has emphasized the rational-number ideas themselves more than the processes for using the ideas. The bulk of our earlier research takes the first approach, whereas the APS project also utilizes the third.

Both the interviews and the written tests focused on fraction and ratio ideas, two of the within-concept networks subsumed by general rational-number understanding, and attended particularly to various representations of these ideas and translations among these representations. The written tests assessed children's abilities to translate within and among paper-and-pencil modes of representation: static figures, written symbols, and written language. The interviews made it possible to assess translations involving concrete materials and more realistic representations, in addition to those mentioned above.

We have investigated other translations in modified testing situations in which the stimulus was not written (either spoken or displayed using concrete materials) but answers were written, and in interviews, in which the stimulus and response could involve spoken language, written symbols and language, and concrete materials (Landau, Hamilton, & Hoy, 1981). The second phase of the RN project is examining the role of spoken language as an intermediary between problem situations and written symbolism (Behr et al., Note 3); the use of pictorial representations as an aid in problem solving is also being investigated (Landau, Note 5). Thus, we have a more general interest in modeling processes, of which paper-and-pencil translations emphasized on the written tests are only one type.

The two components of the conceptual model and the method for building tasks that have been investigated and used in the research reported here provide needed underpinnings for our current work in the APS project, which addresses the remaining two components of the conceptual model within the context of tasks originating in real situations. The between-concept systems of the rational number conceptual model must be addressed in the APS research because, in so many of the larger, applied problems appropriate for our seventh-grade subjects, the ideas–fraction, ratio, proportion, percent–are inextricably connected to what the subjects know about measurement, area, and number lines, as well as to real-world understandings.

When the goal of research is to investigate the processes, skills, and understandings that enable youngsters to use their current understandings of a mathematical idea, it is appropriate to use the kind of small problems discussed in this chapter. However, when the goal is to study the mechanisms by which learners modify and adapt their understandings in the course of solving problems, it becomes more important to focus on the larger kinds of problems that characterize our current applied problem-solving research. The modeling mechanisms students use for creating and refining various interpretations of problem situations will be emphasized in future reports on findings from the APS project.

Reference Notes

1 Behr, M., Lesh, R., & Post, T. The role of manipulative aids in the learning of rational numbers. RISE Grant #SED 79-20591. Northern Illinois University.

2 Lesh, R. Applied problem-solving in middle-school mathematics. RISE grant #SED 80 - 17771. Northwestern University.

3 Behr, M., Lesh, R., & Post, T. The role of representational systems in the acquisition and use of rational number concepts. RISE grant # SED 79-20591. Northern Illinois University.

4 Usiskin, Z. Arithmetic and its applications. RISE grant #SED 79-19065. The University of Chicago.

5 Landau, M. The effect of spatial abilities and problem presentation formats on problem solving performance in middle school students. Doctoral dissertation, in progress. Northwestern University.

References

Cardone, I. P. Centering/decentering and socio-emotional aspects of small groups: An ecological approach to reciprocal relations. Unpublished doctoral dissertation. Northwestern University, 1977.

Kieran, T. E. On the mathematical, cognitive, and instructional foundations of rational numbers. In R. A. Lesh (Ed.), Number and measurement: Papers from a research workshop. Columbus: ERIC/SMEAC, 1976.

Kieran, T. E., & Southwell, B. The development in children and adolescents of the construct of rational numbers as operators. The Alberta Journal of Educational Research, 1979, 25(4), 234-247.

Landau, M., Hamilton, E., & Hoy, C. Relationships between process use and content understanding. Paper presented at the Annual Meeting of the American Educational Research Association, Los Angeles, April 1981.

Lesh, R. Directions for research concerning number and measurement concepts. In R. Lesh (Ed.), Number and measurement: Papers from a research workshop. Columbus: ERIC/SMEAC, 1976.

Lesh, R. Social/affective factors influencing problem solving capabilities. Paper presented at the Third International Conference for the Psychology of Mathematics Education, Warwick, England. 1979. (a)

Lesh, R. Mathematical learning disabilities: Considerations for identification, diagnosis, and remediation. In R. Lesh, E. Mierkiewicz, & M. G. Kantowski (Eds.), Applied mathematical problem solving. Columbus, ERIC/SMEAC, 1979. (B)

Lesh, R. Applied Mathematical Problem Solving. Educational Studies in Mathematics, 1981, 12, 235-264.

Lesh, R. Modeling students' modeling behaviors. In S. Wagner (Ed.), Proceedings of the Fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Athens, Georgia: University of Georgia, 1982.

Lesh R. Metacognition in mathematical problem solving (Tech. Rep.). Evanston, Illinois: Mathematics Learning Research Center, Northwestern University, 1983.

Lesh, R., & Hamilton, E. The rational number testing program. Paper presented at the annual meeting of the American Educational Research Association, Los Angeles, April, 1981.

Lesh, R., Landau, M., & Hamilton, E. Rational number ideas and the role of representational systems. In R. Karplus (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education. Berkeley: Lawrence Hall of Science, 1980.

Lesh, R., & Mierkiewicz, D. Recent research concerning the development of spatial and geometric concepts. Columbus: ERIC/SMEAC, 1978.

*This research was supported in part by the National Science Foundation under grants SED 79-20591 and SED 80-17771. Any opinions, findings, and conclusions expressed in this chapter are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Click here for Appendix 9.A

Click here for Appendix 9.B

You will need Adobe Acrobat Reader to read the appendices.
Click here to go to the Adobe website and to download Acrobat Reader.

(top)